## Bounds for the minimum eigenvalue of a symmetric Toeplitz matrix

Heinrich Voss

### Abstract

In a recent paper Melman [12] derived upper bounds for the smallest eigenvalue of a real symmetric Toeplitz matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation $f(\lambda)=0$, the best of which being constructed by the $(1,2)$-Padé approximation of $f$. In this paper we prove that this bound is the smallest eigenvalue of the projection of the given eigenvalue problem onto a Krylov space of $T_n^{-1}$ of dimension 3. This interpretation of the bound suggests enhanced bounds of increasing accuracy. They can be substantially improved further by exploiting symmetry properties of the principal eigenvector of $T_n$.

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### Key words

Toeplitz matrix, eigenvalue problem, symmetry.

65F15.

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